Suppof. Results

Divifors.

x= 120 1,2,3,4,5,6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 1 20.

1

x=

-10

721, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.

301, 2, 3, 5, 6, 10, 15, 30.

their common difference equal to unit, the first

Of these four arithmetical progreffions having

IO I 23

= gives x 9, the others give x = 2, x = — 3, * = 4; all which fucceed except x=-3: fo that the three values of x are + 9, — 2, — 4.

--

CHA P. VII.

OF THE RESOLUTIONS OF EQUATIONS BY FINDING THE EQUATIONS OF A LOWER DEGREE THAT ARE THEIR DIVISORS.

§ 62. Ta

$62.0 find the roots of an equation is the same thing as to find the simple equal tions, by the multiplication of which into one another it is produced, or to find the fimple equations that divide it without a remainder.

If fuch fimple equations cannot be found, yet if we can find the quadratic equations from which the propofed equation is produced, we may discover its roots afterwards by the refolution of these quadratic equations. Or, if nei. ther these simple equations nor these quadratic equations can be found, yet, by finding a cubic or biquadratic that is a divifor of the propofed equation, we may deprefs it lower, and make the folution more easy.

Now, in order to find the Rules by which thefe divifors may be difcovered, we shall fuppose that

divifors of the propofed equation; and if E represent the quotient arifing by dividing the propofed equation by that divifor, then

Ex mx2- nx + r,

or, Ex mx3

nx2 + rx - s, will reprefent the propofed equation itself. Where it is plain, that "fince m is the coefficient of the highest term of the divifors, it must be a divifor of the coefficient of the highest term of the proposed equation."

§ 63. Next we are to observe, that, supposing the equation has a fimple divifor mx n, if we fubftitute in the equation Ex mxn, in place of x, any quantity, as a, then the quantity that will refult from this fubstitution will neceffarily have man for one of its divifors; fince, in this fubftitution, mx n becomes

If we fubftitute fucceffively for x any arithmetical progreffion, a, a e, a — 2e, &c. the quantities that will refult from thefe fubftitutions will have among their divifors

ma

me- n,

2me. n, which are also in arithmetical progreffion, having their common difference equal to me.

If, for example, we fubftitute for x the terms of this progreffion, 1, 0, 1, the quantities that refult have among their divifors the arithmetical progreffion m―n,

-

n,

m n;

or, changing the figns, n m, n, n + m. Where the difference of the terms is m, and the term belonging to the fuppofition of x = o is n.

§ 64. It is manifest therefore, that when an equation has any fimple divifor, if you fubftitute for x the progreffion 1, 0, -1, there will be found amongst the divifors of the fums that refult from thefe fubftitutions, one arithmetical progreffion at leaft, whofe common difference will be unit or a divifor m of the coefficient of the highest term, and which will be the coefficient of in the fimple divifor required; and whose term, arifing from the supposition of * = 0, will be n the other member of the fimple divifor mx

n.

From which this Rule is deduced for difcovering fuch a fimple divifor, when there is

any,

0 3

RULE.

RULE.

.

"Substitute for x in the propofed equation fucceffively the numbers 1, 0, -1. Find all the divifers of the fums that refult from this fubftitution, and take out all the arithmetical progreffions you can find amongst them, whofe difference is unit, or fome divifor of the coefficient of the highest term of the equation. Then fuppofe n equal to that term of any one progreffion that arifes from the fuppofition of ⇒x=0, and m = the forefaid divifor of the coefficient of the highest term of the equation, which m is alfo the difference of the terms of this progreffion; fo fhall you have mx for the divifor required."

You may find arithmetical progreffions giving divifors that will not fucceed; but if there is any divifor, it will be found thus by means of these arithmetical progreffions.

$65. If the equation propofed has the coefficient of its highest term = 1, then it will be m = 1, and the divifor will be xn, and the rule will coincide with that given in the end of the laft chapter, which we demonstrated after a different manner; for the divifor being x-, the value of x will be + n, the term of the progreffion that is a divifor of the fum that arifes from fuppofing xo. Of this cafe we